Oxygen molecules \(\left(\mathrm{O}_{2}\right)\) are 16 times as massive as hydrogen molecules \(\left(\mathrm{H}_{2}\right)\). Carbon dioxide molecules \(\left(\mathrm{CO}_{2}\right)\) are 22 times as massive as \(\mathrm{H}_2.\) a. Compare the average speed of \(\mathrm{O}_{2}\) and \(\mathrm{CO}_{2}\), molecules in a volume of air. b. Does this ratio depend on air temperature?

To understand how gas molecules behave, it's essential to know that they are constantly in motion. The speed at which gas molecules move can be understood through the kinetic molecular theory. This theory tells us that the average speed of gas molecules depends on their mass. Lighter molecules move faster, while heavier molecules move slower. For example, hydrogen molecules (\text{H}_2) are much lighter compared to oxygen (\text{O}_2) and carbon dioxide (\text{CO}_2) molecules.
The average speed of gas molecules is given by the formula: \( v_{\text{avg}} = \frac{1}{\text{mass}} \). Essentially, as the mass of the molecule increases, the average speed decreases. This relationship is inversely proportional to the square root of the molecule's mass.

The relationship between molecular mass and speed is crucial in understanding gas behavior. The average speed of a gas molecule is inversely proportional to the square root of its mass. This can be mathematically represented as: \( v_{\text{avg}} \text{ is inversely proportional to } \frac{1}{\text{sqrt(mass)}} \).
Let's break this down with an example from the problem:

• Oxygen (\text{O}_2) molecules are 16 times as massive as hydrogen (\text{H}_2) molecules. Using the relationship, the average speed of \text{O}_2 compared to \text{H}_2 is \( \frac{v_{\text{O}_2}}{v_{\text{H}_2}} = \frac{1}{\text{sqrt}(16)} = \frac{1}{4} \).
• On the other hand, carbon dioxide (\text{CO}_2) molecules are 22 times as massive as \text{H}_2 molecules. The average speed ratio is thus \( \frac{v_{\text{CO}_2}}{v_{\text{H}_2}} = \frac{1}{\text{sqrt}(22)} \).

So, when we compare the speeds of \text{O}_2 and \text{CO}_2, we get: \( \frac{v_{\text{O}_2}}{v_{\text{CO}_2}} = \frac{\frac{1}{4}}{\frac{1}{\text{sqrt}(22)}} \). This simplifies to \( \text{sqrt}(22)/4 \), which is approximately equal to a ratio of 1:1.87. This means \text{O}_2 molecules move faster than \text{CO}_2 molecules, even though both are slower than \text{H}_2 molecules.

The speed of gas molecules increases with temperature. This is because higher temperatures provide more energy to the molecules, causing them to move faster. However, the ratio of the speeds of different gas molecules at the same temperature remains constant.
This means that while the absolute speeds of both \text{O}_2 and \text{CO}_2 will increase with rising temperature, their speed ratio (approximately 1:1.87 from earlier calculations) will not change.
The kinetic energy of gas molecules, which drives their speed, is directly proportional to the temperature. Mathematically, this can be represented as: \[ E_{\text{kinetic}} = \frac{3}{2}kT \] where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature.
Thus, if you increase the temperature, the kinetic energy, and consequently the speed of the gas molecules, will increase. However, the ratio remains unaffected because temperature affects all gases equally.